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LinearRecurrence[{4, -7, 8, -7, 4, -1}, {1, 5, 15, 37, 77, 138}, 50] (* Harvey P. Dale, Jun 19 2024 *)
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a(n)+A213486(n)=(n+1)^3.
<a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-7,8,-7,4,-1).
a(n) = (n+1)^3 - A213486(n).
t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[w + x + y >= Abs[w - x] + Abs[x - y] + Abs[y - w], s = s + 1], {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
(Do[If[w + x + y >= Abs[w - x] + Abs[x - y] + Abs[y - w],
s = s + 1], {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
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<a href="/index/Rec">Index to sequences with entries for linear recurrences with constant coefficients</a>, signature (4,-7,8,-7,4,-1).
<a href="/index/Rec#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (4,-7,8,-7,4,-1).
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allocated for Clark KimberlingNumber of (w,x,y) with all terms in {0,...,n} and |w-x|+|x-y|+|y-w| <= w+x+y.
1, 5, 15, 37, 77, 138, 223, 338, 489, 679, 911, 1191, 1525, 1916, 2367, 2884, 3473, 4137, 4879, 5705, 6621, 7630, 8735, 9942, 11257, 12683, 14223, 15883, 17669, 19584, 21631, 23816, 26145, 28621, 31247, 34029, 36973, 40082, 43359, 46810
0,2
<a href="/index/Rec#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (4,-7,8,-7,4,-1).
a(n) = 4*a(n-1)-7*a(n-2)+8*a(n-3)-7*a(n-4)+4*a(n-5)-a(n-6).
G.f.: (1 + x + 2*x^2 + 4*x^3 + x^4)/((1 - x)^4 (1 + x^2)).
t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w + x + y >= Abs[w - x] + Abs[x - y] + Abs[y - w],
s = s + 1], {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]] (* A213487 *)
Cf. A212959.
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nonn,easy
Clark Kimberling, Jun 13 2012
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